1. Technical Field
The present invention is concerned with matched filtering. One example of the use of matched filtering is in imaging applications, where an original signal is input to some system which produces a response and one wishes to characterise the system by comparing the response with the original signal. Some examples of imaging systems are telecommunications line testing and optical time domain reflectometry, where the original signal is transmitted to the line, and the response obtained from the far (or near) end of the line. Another is ranging systems such as RADAR or SONAR where the original signal is transmitted as a radio or ultrasonic wave and the response consists of reflections of it from the surroundings. Others include magnetic recording channel characterisation, and spatial separation (spectroscopy).
2. Related Art
In general, the response may include the original signal, echoes of the original signal, distortion and noise. Conceptually, the characterisation involves examining the correlation between the original signal and the response in order to distinguish those aspects of the response that are a result of the original signal, rather than the result of some other signal, or other interference or noise. Noting that the correlation (or convolution) of two signals in the time domain is equivalent to a multiplication operation in the frequency domain, one can perform this task by multiplying the frequency spectrum of the response by the frequency spectrum of the original signal A common approach to implementation of matched filtering in this way is to form the Fourier transform of the response, multiply it by the Fourier transform of the original signal, and take the inverse Fourier transform of the product. A particularly attractive approach is to deal with signals containing a number of signal samples equal to a power of two, so that the transforms may be performed using the fast Fourier transform (FFT).
Of particular interest is the situation where the original signal consists of a series of pulses. In the analysis given below, such a sequence is considered as a series of Dirac impulses (i.e. idealised pulses of infinitesimal duration) of magnitude +1 or −1 at regular time intervals. In practice, of course the original signal will consist of a band-limited version of such an idealised sequence, For convenience of notation, such a sequence will sometimes be expressed as a binary number—where, for example, a pulse sequence (1, 1, 1, −1) would be represented as 1110.
The Golay complementary sequences are well known (see M. J. E. Golay, “Complementary Series”, IRE Transactions on Information Theory, vol. 7, pp. 82-87, April 1961). They are pairs of finite binary sequences with certain useful autocorrelation properties.
We define the following notation:    X+ is a binary sequence (e.g. 1110); X− is its inverse (e.g. 0001).    The concatenation of two such symbols represents the concatenation of the binary sequences—thus X+X− would, with the example values just given, represent 11100001.    x(t) is the time domain sequence corresponding to X+ (e.g.(+1,+1,+1,−1).    X(f) is the Fourier transform of x(t).
A Golay pair consists of two sequences A+ and B+, each of length N, having the property that the sum of the autocorrelation of a(t) and the autocorrelation of b(t) (both computed at a shift k) is 2N for k=0 and zero for all |k|>0. Examples of Golay pairs having lengths equal to a power of two may be generated by the following iteration formula (written in the binary notation), where a pair in which both sequences have 2K bits is written (A+K,B+K):code of length 20(=1): let (A+0,B+0)=(1,1)code of length 2K+1: let (A+K+1,B+K+1)=(A+KB+K, A+KB−K)For example, (A+0,B+0)=(1,1)(A+1,B+1)=(11,10)(A+2,B+2)=(1110,1101)(A+3,B+3)=(11101101, 11100010)
etc.
This iteration produces only one pair for a particular value of K, whereas, in general, there are more: other examples can be generated from these examples using the procedures described in the Golay paper cited above.